Introduction
xxi
as a sum of four squares by replacing squares with triangular numbers,
obtaining, in our view, a cleaner result. A last example of the type of result
we will discuss in this chapter is the following. Let S be a set consisting
of the positive integers with an additional copy of those positive integers
congruent to zero modulo 7. Decompose S into its even and odd parts, E
and O respectively. Denote by PE{^) and Po(k) the number of partitions
of k with parts taken from the sets E and O respectively. We prove that for
all nonnegative integers fc,
PE(2k) = P0(2k + l).
The prerequisites for this book are a thorough understanding of material
traditionally covered in first year graduate courses - especially the contents
of the complex analysis course. We review, however, the most salient points
about elliptic function theory portions of this course. Although a knowledge
of Riemann surfaces and Fuchsian groups is helpful, it is not needed by the
reader who is willing to accept the summaries of the required material (with
references to the literature). Although we do not, in general, reproduce
material available in other textbooks, we make an exception for material
on theta functions and theta constants despite the availability of excellent
sources (for example, [23]). We do so, not only for the convenience of the
reader, but also to emphasize our point of view. We have also ignored, to a
great extent, the combinatorial and special functions connection. These are
discussed in [2], [7] and [10], for example.
A road map
While we did not intend to write an encyclopedic text, the result has been
quite a large book. We take the liberty of offering the readers our suggestions
for possible ways of going through this text, which was written with several
different types of readers in mind. These range from the beginning grad-
uate mathematics student through the professional mathematician whose
interests are either in combinatorial mathematics (partition theory, repre-
sentation as sums of squares, counting points on conic sections) or function
theory (Riemann surfaces, modular forms). Theoretical physicists might be
interested in portions of the material we cover.
The reader is expected to have a reasonable knowledge of the theory of
functions of a complex variable, through the Riemann mapping theorem,
and enough mathematical maturity to follow an argument even though un-
familiar with the proofs of all the tools used. Thus, the book can be used as
a text for a topics course in either analysis or analytic number theory, and as
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